The following picture presents a simplified scheme
of decision-making.

Here S is a set on environment states,
D is a set of possible decisions, R is a set
of achievable results. Result is influenced by both decision
and environment state. Thus, a mathematical model of the
object just described is a mapping M: S x D --> R,
that for an environment state s and decision d
calculates the result r = M(s,d).

Environment state is usually uncertain. In the framework of risk theory
the uncertainty is described by a probabilistic model, that is, by a
probability distribution on S.
Together with the mapping M this distribution for each decision
d from D induces a distribution on R. Thus for
each decision there is a probability distribution on R, so
making the best decision mean choosing the "best" distribution on R
among those available.

Making optimal decision means choosing the better from the
two probability distributions. A well known approach consists
of assigning an "utility" U(r) to each result r,
calculating expected (mean) utility u of result of each decision
with subsequent choice of decision that leads to greater expected
utility. Now let us assign utilities as described in the following
table:

value

Utility

Awful

0

Bad

2

Good

5

Excellent

10

Calculating expected utilities brings us

u(I) = 0.3 * 2 + 0.7 * 5 = 4.1,

u(O) = 0.3 * 0 + 0.7 * 10 = 7.

Thus expected utility of outdoors picnic in greater
than that of indoors picnic; so we are heading to forest.